Syntax and ParsingSyntax and ParsingCOMS W4115
Prof. Stephen A. EdwardsSpring 2003
Columbia UniversityDepartment of Computer Science
Lexical Analysis(Scanning)
Lexical Analysis(Scanning)
Lexical Analysis (Scanning)Lexical Analysis (Scanning)
Goal is to translate a stream of characters
i n t sp g c d ( i n t sp
a , sp i n t sp b
into a stream of tokens
ID
int
ID
gcd
LPAREN
(
ID
int
ID
a
COMMA
,
ID
int
ID
b
Each token consists of a token type and its text.
Whitespace and comments are discarded.
Lexical AnalysisLexical Analysis
Goal: simplify the job of the parser.
Scanners are usually much faster than parsers.
Discard as many irrelevant details as possible (e.g.,whitespace, comments).
Parser does not care that the the identifer is“supercalifragilisticexpialidocious.”
Parser rules are only concerned with token types.
The ANTLR Compiler GeneratorThe ANTLR Compiler Generator
Language and compiler for writing compilers
Running ANTLR on an ANTLR file produces Java sourcefiles that can be compiled and run.
ANTLR can generate
• Scanners (lexical analyzers)
• Parsers
• Tree walkers
An ANTLR File for a Simple ScannerAn ANTLR File for a Simple Scanner
class CalcLexer extends Lexer;
LPAREN : ’(’ ; // Rules for puctuationRPAREN : ’)’ ;
STAR : ’*’ ;
PLUS : ’+’ ;
SEMI : ’;’ ;
protected // Can only be used as a sub-ruleDIGIT : ’0’..’9’ ; // Any character between 0 and 9INT : (DIGIT)+ ; // One or more digits
WS : (’ ’ | ’\t’ | ’\n’| ’\r’) // Whitespace{ $setType(Token.SKIP); } ; // Action: ignore
ANTLR Specifications for ScannersANTLR Specifications for Scanners
Rules are names starting with a capital letter.
A character in single quotes matches that character.
LPAREN : ’(’ ;
A string in double quotes matches the string
IF : "if" ;
A vertical bar indicates a choice:
OP : ’+’ | ’-’ | ’*’ | ’/’ ;
ANTLR SpecificationsANTLR Specifications
Question mark makes a clause optional.
PERSON : ("wo")? ’m’ (’a’|’e’) ’n’ ;
(Matches man, men, woman, and women.)
Double dots indicate a range of characters:
DIGIT : ’0’..’9’;
Asterisk and plus match “zero or more,” “one or more.”
ID : LETTER (LETTER | DIGIT)* ;
NUMBER : (DIGIT)+ ;
Kleene ClosureKleene Closure
The asterisk operator (*) is called the Kleene Closureoperator after the inventor of regular expressions, StephenCole Kleene, who pronounced his last name “CLAY-nee.”
His son Ken writes “As far as I am aware thispronunciation is incorrect in all known languages. I believethat this novel pronunciation was invented by my father.”
Scanner BehaviorScanner Behavior
All rules (tokens) are considered simultaneously. Thelongest one that matches wins:
1. Look at the next character in the file.
2. Can the next character be added to any of the tokensunder construction?
3. If so, add the character to the token being constructedand go to step 1.
4. Otherwise, return the token.
How to keep track of multiple rules matchingsimultaneously? Build an automata.
Implementing ScannersAutomaticallyImplementing ScannersAutomatically
Regular Expressions (Rules)
Nondeterministic Finite Automata
Subset Construction
Deterministic Finite Automata
Tables
Regular Expressions and NFAsRegular Expressions and NFAs
We are describing tokens with regular expressions:
• The symbol ε always matches
• A symbol from an alphabet, e.g., a, matches itself
• A sequence of two regular expressions e.g., e1e2
Matches e1 followed by e2
• An “OR” of two regular expressions e.g., e1|e2
Matches e1 or e2
• The Kleene closure of a regular expression, e.g., (e)∗
Matches zero or more instances of e1 in sequence.
Deterministic Finite AutomataDeterministic Finite Automata
A state machine with an initial state
Arcs indicate “consumed” input symbols.
States with double lines are accepting.
If the next token has an arc, follow the arc.
If the next token has no arc and the state is accepting,return the token.
If the next token has no arc and the state is not accepting,syntax error.
Deterministic Finite AutomataDeterministic Finite Automata
ELSE: "else" ;
ELSEIF: "elseif" ;
e l s e
i
f
Deterministic Finite AutomataDeterministic Finite Automata
IF: "if" ;
ID: ’a’..’z’ (’a’..’z’ | ’0’..’9’)* ;
NUM: (’0’..’9’)+ ;
ID IF
ID ID
NUM NUM
i
f
a-z0-9
a-eg-z0-9
a-z90-9a-hj-z a-z0-9
0-9
0-90-9
Nondeterminstic Finite AutomataNondeterminstic Finite Automata
DFAs with ε arcs.
Conceptually, ε arcs denote state equivalence.
ε arcs add the ability to make nondeterministic(schizophrenic) choices.
When an NFA reaches a state with an ε arc, it moves intoevery destination.
NFAs can be in multiple states at once.
Translating REs into NFAsTranslating REs into NFAs
aa
e1e2 e1 e2
ε
e1|e2
e1
e2
ε
ε
ε
ε
(e)∗ eε
ε
ε
ε
RE to NFAsRE to NFAs
Building an NFA for the regular expression
(wo|ε)m(a|e)n
produces
w o
ε
ma
e
n
after simplification. Most ε arcs disappear.
Subset ConstructionSubset Construction
How to compute a DFA from an NFA.
Basic idea: each state of the DFA is a marking of the NFA
w
o
m
m
a
en
Subset ConstructionSubset Construction
An DFA can be exponentially larger than thecorresponding NFA.
n states versus 2n
Tools often try to strike a balance between the tworepresentations.
ANTLR uses a different technique.
Free-Format LanguagesFree-Format Languages
Typical style arising from scanner/parser division
Program text is a series of tokens possibly separated bywhitespace and comments, which are both ignored.
• keywords (if while)
• punctuation (, ( +)
• identifiers (foo bar)
• numbers (10 -3.14159e+32)
• strings ("A String")
Free-Format LanguagesFree-Format Languages
Java C C++ Algol Pascal
Some deviate a little (e.g., C and C++ have a separatepreprocessor)
But not all languages are free-format.
FORTRAN 77FORTRAN 77
FORTRAN 77 is not free-format. 72-character lines:
100 IF(IN .EQ. ’Y’ .OR. IN .EQ. ’y’ .OR.
$ IN .EQ. ’T’ .OR. IN .EQ. ’t’) THEN
1 · · · 5︸ ︷︷ ︸
Statement label
6︸︷︷︸
Continuation
7 · · · 72︸ ︷︷ ︸
Normal
When column 6 is not a space, line is considered part ofthe previous.
Fixed-length line works well with a one-line buffer.
Makes sense on punch cards.
PythonPython
The Python scripting language groups with indentation
i = 0
while i < 10:
i = i + 1
print i # Prints 1, 2, ..., 10
i = 0
while i < 10:
i = i + 1
print i # Just prints 10
This is succinct, but can be error-prone.
How do you wrap a conditional around instructions?
Syntax and Langauge DesignSyntax and Langauge Design
Does syntax matter? Yes and no
More important is a language’s semantics—its meaning.
The syntax is aesthetic, but can be a religious issue.
But aesthetics matter to people, and can be critical.
Verbosity does matter: smaller is usually better.
Too small can be a problem: APL is a compact, crypticlanguage with its own character set (!)
E←A TEST B;L
L←0.5
E←((A×A)+B×B)*L
Syntax and Language DesignSyntax and Language Design
Some syntax is error-prone. Classic FORTRAN example:
DO 5 I = 1,25 ! Loop header (for i = 1 to 25)
DO 5 I = 1.25 ! Assignment to variable DO5I
Trying too hard to reuse existing syntax in C++:
vector< vector<int> > foo;
vector<vector<int>> foo; // Syntax error
C distinguishes > and >> as different operators.
KeywordsKeywords
Keywords look like identifiers in most languages.
Scanners do not know context, so keywords must takeprecedence over identifiers.
Too many keywords leaves fewer options for identifiers.
Langauges such as C++ or Java strive for fewer keywordsto avoid “polluting” available identifiers.
ParsingParsing
ParsingParsing
Objective: build an abstract syntax tree (AST) for thetoken sequence from the scanner.
2 * 3 + 4 ⇒
+
*
2 3
4
Goal: discard irrelevant information to make it easier forthe next stage.
Parentheses and most other forms of punctuationremoved.
GrammarsGrammars
Most programming languages described using acontext-free grammar.
Compared to regular languages, context-free languagesadd one important thing: recursion.
Recursion allows you to count, e.g., to match pairs ofnested parentheses.
Which languages do humans speak? I’d say it’s regular: Ido not not not not not not not not not not understand thissentence.
LanguagesLanguages
Regular languages (t is a terminal):
A→ t1 . . . tnB
A→ t1 . . . tn
Context-free languages (P is terminal or a variable):
A→ P1 . . . Pn
Context-sensitive languages:
α1Aα2 → α1Bα2
“B → A only in the ‘context’ of α1 · · ·α2”
IssuesIssues
Ambiguous grammars
Precedence of operators
Left- versus right-recursive
Top-down vs. bottom-up parsers
Parse Tree vs. Abstract Syntax Tree
Ambiguous GrammarsAmbiguous Grammars
A grammar can easily be ambiguous. Consider parsing
3 - 4 * 2 + 5
with the grammar
e→ e + e | e− e | e ∗ e | e / e
+
-
3 *
4 2
5
-
3 +
*
4 2
5
*
-
3 4
+
2 5
-
3 *
4 +
2 5
+
*
-
3 4
2
5
Operator Precedence andAssociativityOperator Precedence andAssociativity
Usually resolve ambiguity in arithmetic expressions
Like you were taught in elementary school:
“My Dear Aunt Sally”
Mnemonic for multiplication and division before additionand subtraction.
Operator PrecedenceOperator Precedence
Defines how “sticky” an operator is.
1 * 2 + 3 * 4
* at higher precedence than +:(1 * 2) + (3 * 4)
+
*
1 2
*
3 4
+ at higher precedence than *:1 * (2 + 3) * 4
*
*
1 +
2 3
4
AssociativityAssociativity
Whether to evaluate left-to-right or right-to-left
Most operators are left-associative
1 - 2 - 3 - 4
-
-
-
1 2
3
4
-
1 -
2 -
3 4((1 - 2) - 3) - 4 1 - (2 - (3 - 4))
left associative right associative
Fixing Ambiguous GrammarsFixing Ambiguous Grammars
Original ANTLR grammar specification
expr
: expr ’+’ expr
| expr ’-’ expr
| expr ’*’ expr
| expr ’/’ expr
| NUMBER
;
Ambiguous: no precedence or associativity.
Assigning Precedence LevelsAssigning Precedence Levels
Split into multiple rules, one per level
expr : expr ’+’ expr
| expr ’-’ expr
| term ;
term : term ’*’ term
| term ’/’ term
| atom ;
atom : NUMBER ;
Still ambiguous: associativity not defined
Assigning AssociativityAssigning Associativity
Make one side or the other the next level of precedence
expr : expr ’+’ term
| expr ’-’ term
| term ;
term : term ’*’ atom
| term ’/’ atom
| atom ;
atom : NUMBER ;
Parsing Context-Free GrammarsParsing Context-Free Grammars
There are O(n3) algorithms for parsing arbitrary CFGs,but most compilers demand O(n) algorithms.
Fortunately, the LL and LR subclasses of CFGs haveO(n) parsing algorithms. People use these in practice.
Parsing LL(k) GrammarsParsing LL(k) Grammars
LL: Left-to-right, Left-most derivation
k: number of tokens to look ahead
Parsed by top-down, predictive, recursive parsers
Basic idea: look at the next token to predict whichproduction to use
ANTLR builds recursive LL(k) parsers
Almost a direct translation from the grammar.
A Top-Down ParserA Top-Down Parser
stmt : ’if’ expr ’then’ expr
| ’while’ expr ’do’ expr
| expr ’:=’ expr ;
expr : NUMBER | ’(’ expr ’)’ ;
AST stmt() {switch (next-token) {case ”if” : match(”if”); expr(); match(”then”); expr();case ”while” : match(”while”); expr(); match(”do”); expr();case NUMBER or ”(” : expr(); match(”:=”); expr();}
}
Writing LL(k) GrammarsWriting LL(k) Grammars
Cannot have left-recursion
expr : expr ’+’ term | term ;
becomes
AST expr() –switch (next-token) –case NUMBER : expr(); /* Infinite Recursion */
Writing LL(1) GrammarsWriting LL(1) Grammars
Cannot have common prefixes
expr : ID ’(’ expr ’)’
| ID ’=’ expr
becomes
AST expr() –switch (next-token) –case ID : match(ID); match(’(’); expr(); match(’)’);case ID : match(ID); match(’=’); expr();
Eliminating Common PrefixesEliminating Common Prefixes
Consolidate common prefixes:
expr
: expr ’+’ term
| expr ’-’ term
| term
;
becomes
expr
: expr (’+’ term | ’-’ term )
| term
;
Eliminating Left RecursionEliminating Left Recursion
Understand the recursion and add tail rules
expr
: expr (’+’ term | ’-’ term )
| term
;
becomes
expr : term exprt ;
exprt : ’+’ term exprt
| ’-’ term exprt
| /* nothing */
;
Using ANTLR’s EBNFUsing ANTLR’s EBNF
ANTLR makes this easier since it supports * and -:
expr : expr ’+’ term
| expr ’-’ term
| term ;
becomes
expr : term (’+’ term | ’-’ term)* ;
The Dangling Else ProblemThe Dangling Else Problem
Who owns the else?
if (a) if (b) c(); else d();
if
a if
b c() d()
or if
a if
b c()
d()
?
Grammars are usually ambiguous; manuals givedisambiguating rules such as C’s:
As usual the “else” is resolved by connecting anelse with the last encountered elseless if.
The Dangling Else ProblemThe Dangling Else Problem
stmt : "if" expr "then" stmt iftail
| other-statements ;
iftail
: "else" stmt
| /* nothing */
;
Problem comes when matching “iftail.”
Normally, an empty choice is taken if the next token is inthe “follow set” of the rule. But since “else” can follow aniftail, the decision is ambiguous.
The Dangling Else ProblemThe Dangling Else Problem
ANTLR can resolve this problem by making certain rules“greedy.” If a conditional is marked as greedy, it will takethat option even if the “nothing” option would also match:
stmt
: "if" expr "then" stmt
( options {greedy = true;}
: "else" stmt
)?
| other-statements
;
The Dangling Else ProblemThe Dangling Else Problem
Some languages resolve this problem by insisting onnesting everything.
E.g., Algol 68:
if a < b then a else b fi;
“fi” is “if” spelled backwards. The language also usesdo–od and case–esac.
Statement separators/terminatorsStatement separators/terminators
C uses ; as a statement terminator.
if (a<b) printf("a less");
else {
printf("b"); printf(" less");
}
Pascal uses ; as a statement separator.
if a < b then writeln(’a less’)
else begin
write(’a’); writeln(’ less’)
end
Pascal later made a final ; optional.
Bottom-up ParsingBottom-up Parsing
Rightmost DerivationRightmost Derivation
1 : e→t + e
2 : e→t
3 : t→Id ∗ t
4 : t→Id
A rightmost derivation for Id ∗ Id + Id:
e
t + e
t + t
t + IdId ∗ t + IdId ∗ Id + Id
Basic idea of bottom-up parsing:construct this rightmost derivationbackward.
HandlesHandles
1 : e→t + e
2 : e→t
3 : t→Id ∗ t
4 : t→Id
Id ∗ Id + Id
Id ∗ t + Id
t + Id
t + t
t + e
e e
t
Id * t
Id
+ e
t
Id
This is a reverse rightmost derivation for Id ∗ Id + Id.
Each highlighted section is a handle.
Taken in order, the handles build the tree from the leavesto the root.
Shift-reduce ParsingShift-reduce Parsing
1 : e→t + e
2 : e→t
3 : t→Id ∗ t
4 : t→Id
stack input actionId ∗ Id + Id shift
Id ∗ Id + Id shiftId∗ Id + Id shiftId ∗ Id + Id reduce (4)Id ∗ t + Id reduce (3)t + Id shiftt+ Id shiftt + Id reduce (4)t + t reduce (2)t + e reduce (1)e accept
Scan input left-to-right, looking for handles.
An oracle tells what to do
LR ParsingLR Parsing
1 : e→t + e
2 : e→t
3 : t→Id ∗ t
4 : t→Idaction goto
Id + ∗ $ e t
0 s1 7 21 r4 r4 s3 r42 r2 s4 r2 r23 s1 54 s1 6 25 r3 r3 r3 r36 r1 r1 r1 r17 acc
stack input action
0Id * Id + Id $ shift, goto 1
1. Look at state on top of stack
2. and the next input token
3. to find the next action
4. In this case, shift the tokenonto the stack and go tostate 1.
LR ParsingLR Parsing
1 : e→t + e
2 : e→t
3 : t→Id ∗ t
4 : t→Idaction goto
Id + ∗ $ e t
0 s1 7 21 r4 r4 s3 r42 r2 s4 r2 r23 s1 54 s1 6 25 r3 r3 r3 r36 r1 r1 r1 r17 acc
stack input action
0Id * Id + Id $ shift, goto 1
0Id1
* Id + Id $ shift, goto 3
0Id1
*3
Id + Id $ shift, goto 1
0Id1
*3
Id1
+ Id $ reduce w/ 4
Action is reduce with rule 4 (t →Id). The right side is removed fromthe stack to reveal state 3. Thegoto table in state 3 tells us to goto state 5 when we reduce a t:
stack input action
0Id1
*3
t
5+ Id $
LR ParsingLR Parsing
1 : e→t + e
2 : e→t
3 : t→Id ∗ t
4 : t→Idaction goto
Id + ∗ $ e t
0 s1 7 21 r4 r4 s3 r42 r2 s4 r2 r23 s1 54 s1 6 25 r3 r3 r3 r36 r1 r1 r1 r17 acc
stack input action
0Id * Id + Id $ shift, goto 1
0Id1
* Id + Id $ shift, goto 3
0Id1
*3
Id + Id $ shift, goto 1
0Id1
*3
Id1
+ Id $ reduce w/ 4
0Id1
*3
t
5+ Id $ reduce w/ 3
0t2
+ Id $ shift, goto 4
0t2
+4
Id $ shift, goto 1
0t2
+4
Id1
$ reduce w/ 4
0t2
+4
t
2$ reduce w/ 2
0t2
+4
e
6$ reduce w/ 1
0e7
$ accept
Constructing the SLR Parse TableConstructing the SLR Parse Table
The states are places we could be in a reverse-rightmostderivation. Let’s represent such a place with a dot.
1 : e→t + e
2 : e→t
3 : t→Id ∗ t
4 : t→Id
Say we were at the beginning (·e). This corresponds to
e′ → ·ee→ ·t + ee→ ·tt→ ·Id ∗ tt→ ·Id
The first is a placeholder. Thesecond are the two possibilitieswhen we’re just before e. The lasttwo are the two possibilities whenwe’re just before t.
Constructing the SLR Parsing TableConstructing the SLR Parsing Table
S7: e′→ e·
S0:
e′→ ·e
e → ·t + e
e → ·t
t → ·Id ∗ t
t → ·Id
S2: e → t · +e
e → t ·S4:
e → t + ·e
e → ·t + e
e → ·t
t → ·Id ∗ t
t → ·Id
S1: t → Id · ∗t
t → Id· S6: e → t + e·
S3:t → Id ∗ ·t
t → ·Id ∗ t
t → ·IdS5: t → Id ∗ t·
Id + ∗ $ et
0 s1 721 r4 r4 s3 r42 r2 s4 r2 r23 s1 54 s1 625 r3 r3 r3 r36 r1 r1 r1 r17 acc
Id
e
t
∗
+
Idt
e
t
Id
The PunchlineThe Punchline
This is a tricky, but mechanical procedure. The parsergenerators YACC, Bison, Cup, and others (but notANTLR) use a modified version of this technique togenerate fast bottom-up parsers.
You need to understand it to comprehend error messages:
Shift/reduce conflicts arecaused by a state like
t→ Id · ∗t
t→ Id ∗ t·
Reduce/reduce conflicts arecaused by a state like
t→ Id ∗ t·
e→ t + e·